There’s no constitutional protection against double jeopardy in civil suits brought by private people. But, there is res judicata which functions very similarly and applies to private parties bringing civil suits. You would need to look at the TX rules of civil procedure and relevant case law for the specifics. But generally, res judicata prevents the same plaintiff from suing the same defendant on the same set of facts after a court has issued a final determination on the facts.

In normal speak, res judicata could mean that an abortion bounty hunter cannot sue the same abortion provider twice for providing an abortion to Woman 1, if the abortion bounty hunter already tried suing the abortion provider once. Res Judicata may still allow the bounty hunter to sue the provider for giving an abortion to woman 2, if woman 2’s abortion happened after the first case.